Large Deviation For Parameter Estimation In Linear Fractional Diffusion Process

Stochastic differential equation plays an important role in Stochastic Calculus, and it has important applications in non-mathematics fields such as financial decision-making, machine control and system biology. In order to solve practical problem, it is essential to consider the statistical inference theory of stochastic differential equations for us.This article mainly discussed the parameter estimation and large deviation for stochastic differential equation by fractional Brownian motion. The similar question in simpler fractional O-U process has studied in many literatures. So, this paper further promoted the result in fractional O-U process, and mainly considered the estimation ofθand large deviation for linear fractional diffusion process.When discussing the question, firstly, this article introduced fractional Brownian motion definition and a lot of properties about the fractional Brownian motion and stochastic integral with respect to the fBM. And then, the paper had defined the non-random function with respect to the fBM using Girsanov type formula for fractional Brownian motion in the chapter 2. Secondly, this article had given the specific form of MLE ofθin the drift coefficient for linear fractional diffusion process using the Radon-Nikodym derivations in the chapter 3. Finally, in the chapter 4, this article had proved two lemmas, and given the large deviation inequality based on other literatures.Stochastic differential equation has its important applications in many non-mathematics fields. Moreover, when we dealt with practical problems, the unknown parameter often appeared in the mathematic model we had built. According to the observed sample path of the random process and the corresponding data, we are able to estimate the parameter and discuss the large deviation, which is our primary problem.